Imaging of cancer with combined positron emission tomography/computed tomography (PET/CT) scanners has become a standard component of oncology diagnosis and staging. Furthermore, quantitative PET/CT is a valuable tool for assessment of an individual's response to therapy and for clinical trials of novel cancer therapies. PET/CT imaging of the lung and abdomen, however, is generally affected by patient respiratory motion, which can lead to underestimation of tracer concentration within a region of interest, overestimation of tumor volume, and mis-matched PET and CT images that yield attenuation correction errors, registration errors and tumor mis-localization. The first source of error (attenuation mismatch) can be addressed by performing CT-based attenuation correction using respiratory gated CT images that are phase-matched with the respiratory gated PET scans. The second source of error (motion blurring) can be addressed by methods using respiratory motion estimation and/or gating. The problem is that even with the lowest possible CT technique, the radiation dose is unacceptably high for diagnostic imaging procedures. Our goal is to reduce CT radiation dose dramatically, while improving PET image by reducing errors introduced by respiratory motion. This will enable more accurate PET/CT imaging by enabling compensation of respiratory motion induced artifacts. This is goal is feasible due to the requirements for the CT component of PET/CT imaging, which are different than those for diagnostic CT. The methods proposed here will (1) reduce radiation dose from CT-based attenuation correction methods and (2) provide routine and high quality compensation of respiratory motion artifacts in PET/CT imaging. This will improve the capabilities of quantitative PET/CT imaging in the development of badly needed therapies for cancer, in the evaluation of response to therapy by for an individual patient, and where lesion uptake is noted is a clinical report. The impact of respiratory motion for detection, a primary tool in diagnosis and staging is not clear, but has also not been properly evaluated yet. In addition, the ultra low dose CT methods developed here may be useful for other applications, such as pulmonary CT imaging, dynamic CT with contrast or new tracers, PET/CT guided radiation treatment planning, and PET/CT cardiac imaging.